Solve 1/2x^2+98+48x=0

$$\frac { 1 } { 2 } x ^ { 2 } + 98 + 48 x = 0$$

Solve for x

$$x=2\sqrt{527}-48\approx -2.087038867$$

$$x=-2\sqrt{527}-48\approx -93.912961133$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$\frac{1}{2}x^{2}+48x+98=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, 48 for b, and 98 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-48±\sqrt{48^{2}-4\times \frac{1}{2}\times 98}}{2\times \frac{1}{2}}$$

Square 48.

$$x=\frac{-48±\sqrt{2304-4\times \frac{1}{2}\times 98}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$x=\frac{-48±\sqrt{2304-2\times 98}}{2\times \frac{1}{2}}$$

Multiply -2 times 98.

$$x=\frac{-48±\sqrt{2304-196}}{2\times \frac{1}{2}}$$

Add 2304 to -196.

$$x=\frac{-48±\sqrt{2108}}{2\times \frac{1}{2}}$$

Take the square root of 2108.

$$x=\frac{-48±2\sqrt{527}}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$x=\frac{-48±2\sqrt{527}}{1}$$

Now solve the equation x=\frac{-48±2\sqrt{527}}{1} when ± is plus. Add -48 to 2\sqrt{527}.

$$x=\frac{2\sqrt{527}-48}{1}$$

Divide -48+2\sqrt{527} by 1.

$$x=2\sqrt{527}-48$$

Now solve the equation x=\frac{-48±2\sqrt{527}}{1} when ± is minus. Subtract 2\sqrt{527} from -48.

$$x=\frac{-2\sqrt{527}-48}{1}$$

Divide -48-2\sqrt{527} by 1.

$$x=-2\sqrt{527}-48$$

The equation is now solved.

$$x=2\sqrt{527}-48 x=-2\sqrt{527}-48$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$\frac{1}{2}x^{2}+48x+98=0$$

Subtract 98 from both sides of the equation.

$$\frac{1}{2}x^{2}+48x+98-98=-98$$

Subtracting 98 from itself leaves 0.

$$\frac{1}{2}x^{2}+48x=-98$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}x^{2}+48x}{\frac{1}{2}}=-\frac{98}{\frac{1}{2}}$$

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

$$x^{2}+\frac{48}{\frac{1}{2}}x=-\frac{98}{\frac{1}{2}}$$

Divide 48 by \frac{1}{2} by multiplying 48 by the reciprocal of \frac{1}{2}.

$$x^{2}+96x=-\frac{98}{\frac{1}{2}}$$

Divide -98 by \frac{1}{2} by multiplying -98 by the reciprocal of \frac{1}{2}.

$$x^{2}+96x=-196$$

Divide 96, the coefficient of the x term, by 2 to get 48. Then add the square of 48 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+96x+48^{2}=-196+48^{2}$$

Square 48.

$$x^{2}+96x+2304=-196+2304$$

Add -196 to 2304.

$$x^{2}+96x+2304=2108$$

Factor x^{2}+96x+2304. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+48\right)^{2}=2108$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+48\right)^{2}}=\sqrt{2108}$$

Simplify.

$$x+48=2\sqrt{527} x+48=-2\sqrt{527}$$

Subtract 48 from both sides of the equation.

$$x=2\sqrt{527}-48 x=-2\sqrt{527}-48$$